_ 3 ^ 2 3 x 1+ x ~d x=

Step-by-step solution

Given the integral I = _ 3 ^ 2 3 x 1+ x , dx , apply the transformation property _a^b f(x) , dx = _a^b f(a+b-x) , dx with a+b = : I = _ 3 ^ 2 3 -x 1+ x , dx Adding this to the original expression yields: 2I = _ 3 ^ 2 3 1+ x , dx Rewriting the integrand through rationalization: 1 1+ x = 1- x ^2 x = ^2 x - x x Integration gives: ( ^2 x - x x ) , dx = x - x + C Evaluating the definite integral: 2I = [ x - x ]_ 3 ^ 2 3 ( 2 3 ) - ( 2 3 ) = - 3 - (-2) = 2- 3 ( 3 ) - ( 3 ) = 3 - 2 2I = [ (2- 3 ) - ( 3 -2) ] = (4 - 2 3 ) =